the Air Vent

Because the world needs another opinion

Fixing Briffa’s Latest

Posted by Jeff Id on November 1, 2009

This is a very difficult post for me because I don’t believe any of this has anything to do with temperature, however some well paid scientists disagree. This is therefore in response to a Keith Briffa pre-paper replying to SteveM’s post on Yamal where he replaced some of the data using a Schweingruber dataset. Briffa presented what he referred to as a sensitivity test of Yamal by adding different datasets in. While he didn’t provide his code, he did provide the data and plots of the RCS corrections applied to the various datasets.

RCS is a generic method of fitting a curve to a set of tree ring data. In the original Yamal an exponential decay was used to represent tree ring widths. I’m critical of the exponential function method because it doesn’t recognize the potential for trees to increase in growth rate as they age. Since nothing I’ve read biologically defines that this is impossible it makes no sense to ignore the possibility unless you want older trees to curve upward after standardization. It’s odd to see it being ignored by the pros but it happens to result in spurious hockey sticks through a complex and surprisingly common mathematical phenomena referred to here as hockestickization.

Yamal is a driver amongst hockey sticks – I mean a good old fashion 1 wood. Straight shaft, sharp blade, wide sweet spot. Since Steve’s discovery the climatoknowledgists have been scrambling to explain how so few trees could define such strong warming around the globe. In Briffa’s response he showed several methods using new data to also create hockey sticks. However, Dr. Briffa again used a method which is functionally identical to the exponential decay method for correcting tree ring widths. It seems he simply cannot recognize that trees may increase in ring width.

Individual site RCS incorporating sub-fossil data

Figure D from Briffa's response. Click to expand. This is a series of chronologies included with Yamal to show that you can always get a similar hockey stick.

You can see that the bottom pane has a variety of curves which all have an uptrend similar to the original Yamal. Briffa did one nice thing in that he didn’t repeat Dr. Tom P’s mistake of including Yamal only years in the data. If the sensitivity data ends early, Briffa correctly ended the comparison curves early. When this story first broke, Tom P argued that Steve McIntyre had failed to do his sensitivity analysis correctly but in fact Tom had mistakenly extended the Yamal data past the sensitivity data. My reply was here. I never did read an admission of error from him on that one.

Anyway, note the curves used for correcting the datasets. They are very similar to the exponential curves used to create the original Yamal emulated in this next figure. Before I continue, much of this post was created from code written and published by Steve McIntyre although I’ve made many changes so mistakes are my own.

yamalex

Fig 2

The last pane of the above figure shows the Yamal hockey stick, the middle pane shows that not many trees were used in recent years to make it. My problem with it is addressed here and that is the exponential curve in the top pane of the above 3 graphs. The curve doesn’t follow the average tree age which trends upward after about 200 years. Since the middle of the series has both young and old trees overlapping, standardization problems average out. This means that errors in standardization will only have a real effect on either end. The problem for RCS is therefore summarized like this:

1 – Standardization affects only ends of chronologies

2 – Trees grow at a variable average rate even in the same climate conditions. This rate is manifested in variable ring widths.

3 – Trees often grow faster after a certain age when competition is beaten and resources are monopolized.

4 – Old trees are easier to find when alive, fossil and subfossil records will tend toward a younger age.

5 – Exponential decay does not correct for increasing growth in later years.

The result then is an uptick at the end of the RCS chronology created by the math. So this post attempts to fix this one aspect of the Yamal series. There are a bunch of graphs to show but the point is pretty simple. I used a spline correction similar to the Esper method, fitting the data to the mean ring width for the various new trees from Briffa.

live

Fig 3

jah

Fig 4

por

Fig 5

russ

Fig 6

yamalsp

Fig 7

OK, now the RUSS series is the Schweingruber series that SteveM used in the Yamal data. Instead of playing around with a boring sensitivity test I combined all of the data into full chronologies a couple of different ways using the maximum amount of data. Before we move on, look at the age of the trees in the first pane of each chronology. Several follow a nice exponential drop until about 200 years followed by a rise later on. There are a couple who don’t follow a nice curve like that, I’m not sure why but the point is that RCS needs the ability to fit the rise and a pure exponential decay doesn’t allow for that. A second issue you’ll notice is that older trees have a lot of variation in ring width. The signal goes wild as trees age. This could be due to less old trees in the series but it is a noticeable effect.
First, I looked at an average of the above series after correction by spline.

allmean

Fig 8

The red line is the new chronology. As I explained above the only real differences created by different RCS methods occur at the ends of the chronology.The huge spike at the end is reduced from the original Yamal but is still positive. The next graph is a zoomed in version which if it were temperature, the red line would probably be a better match to measured data than the original black. However, IMO it isn’t temperature.

allmeans

Fig 9

The next graph is the same data as the above with a 5 year filter instead of a 21 year. It’s almost always better to look at endpoints with less filtering because the ends are affected by the assumptions. So, for those who see temperature in these things, what about this?

allmeans3

Fig 10

We have just removed any resemblance to the hockey sticks from from Yamal by using all the data. This isn’t any less valid than Briffa’s versions.

But wait, I’ve got another version. The above reconstruction uses a different RCS spline fit for each region, there certainly is an argument for that method but the resulting average isn’t weighted according to the number of trees so a series with 10 trees has the same weight as one with 25. I took all the data together and did a single RCS spline fit.

yamalall

Fig 11

The reconstruction is the bottom pane. Great Jeff you made your own hockey stick, nice work right? You can see the huge core count in recent years right before the blade happens. The blade in this case though deserves a closer look.

This is the unfiltered version.

unfyamalall

Fig 12

You can see the blade is very thin in reality. The whole series ends in 1996 so I chopped one year (ONE SINGLE VALUE ON THE END) to see the difference.

unfyamalall1995

Fig 13

Now visually that’s a significant difference from the previous pane yet it’s only chopping a single value from 1996. There were 19 cores in 1996 but in 1994 there were 42. I’m certain that the lack of new and young trees has a lot to do with the unprecedented nature of the spike. You can just look at the lack of variance in the early portion of the curve fit’s (pane 1) in the above chronologies (fig 3, 4,5,6 …) and it becomes very apparent that it would be almost 100% impossible for a group of young trees to make a 3 sigma deviation from the mean of the curve.

I hope that makes sense, it’s what drives me nuts about this treemometer stuff. It just ain’t science.

It aint!!

Anyway, our eyes tend to focus on the big bladey looking bit at the end without realizing just how narrow it really is. Here’s a filtered versioin truncated only 5 years to 1990. I picked 1990 because that’s the year that the data availability jumped up to over 80 cores.

unfyamalall1990

Fig 14

The entire 19th century using all the data is now one of the coolest on record!!

My conclusion is that there is no REAL uptick in Yamal. There are methods which show an uptick but the number of cores is still low. For the uptick to be continuous for 100ish years as corrected HadCRUT temperature is you need to have an RCS which creates a continuous upslope like exponential RCS. What bugs me so much about that is that this continuous slope is just about all that is required for correlation to temp.

No matter how you see the last 3 graphs, they sure as heck don’t look like temp. It’s my contention that they are of equal quality to Briffa’s original Yamal. This post took a long time because there were hundreds of other plots I could have shown and have done, but there are enough plots here for the serious to figure out what’s going on.


28 Responses to “Fixing Briffa’s Latest”

  1. John F. Pittman said

    The 21 year filter does seem to match certain characteristic of temperature, such as between 1810 and 1820, two volcanoes and a minimum affected temperature. The 1930’s had a warm period in the American midwest and the arctic. A mid 20th century cooling period with warming. And it supports Ross’s contention of temperature data contamination starting around 1950. Quick, call a press conference before someone takes your data, and won’t give you credit. Your name may be as verboten as Steve’s.😉

  2. pyromancer76 said

    Jeff Id, thanks for such quick response to Briffa’s new rationalization of bad (if not fraudulent) science.

    However, I have a problem with the organization of your site. Your center column on my computer is narrow. It contains all the text, but I have to click on EVERY chart to get it on a full page. Is there any cure for the problem?

  3. curious said

    Jeff – I’m nowhere near the leading edge of this stuff – (your comment on another thread about “brains and ears” (worth a hit on the tip jar!) sort of sums up my views at the moment and I’m not putting the time in to do the work to get to the bottom of what seems to be a bottomless pit) – but really; is it worth it?

    Apologies if this shows my lack of understanding but what does a spline really do? Is n’t it simply another curve fit? No analytical or predictive power? My knowledge of them is from CAD and drafting where they are great tools and, whilst I’m rusty on the maths, they really don’t do anything more than a curvature controlled fit to (within) data points – or am I missing something? Tom P referenced a Briffa (I think) paper over at CA where he talked about “time variable stiffness” and Bender eventually tracked down a bit of FORTRAN code that was supposed to generate the spline. Again I’m rusty but from my quick skim it looked like it hinged on an external variable (swp?) which had to be supplied to the routine. So what are the criteria for this “time varying stiffnes” variable? Anything related to a causal/physical relationship or simply another data processing step to improve the fit? IMO its as subjective as putting more knots/control points on a curve/surface to improve the look of it or the way the light plays on it. I asked Tom P to elaborate but didn’t hear anymore. If I’m way off please tell me.

    Moving on to the underlying data, MrPete put up a great screen shot at CA from winDENDRO showing a tree slice scan being marked up. This gave me a link to the winDENDRO site where they have a brochure with the product specs and details of the scanners they use. My interest in this was from having read a dendro paper where the accuracy of measurement or ring width was quoted to be 0.001mm – this seemed pretty amazing to me for a biological specimen such as a non circular/concentric tree section sample. Having looked a bit closer it looks as if the absolute best they can do is to get to 0.02mm and that is based on 4 pixels of data – so its pretty much identifying the presence of a ring. Their latest file format does go to 3dp but I think that is just a placeholder. Ryan O mentioned gauge analysis and, yes it would be good, but really is TRW the right variable even? Given the scanning approach you’d think an area based measure would be pretty straight forwards and this could accomodate the issues of variable geometric layout. And that would still need the causal relation to be propoperly identified.

    Sorry this is heading off into rant territory so it’s not well strucutured but really with the work so many in blogland are doing on this how can these crazy half baked arguments still be on the table? IMO it’s time for a clean sheet on this starting with the instrumental records and the historical accounts of geographical climate found in avenues of history and geography. And I know this is happening but not in the mainstream where all we hear is one extreme view after another.

    And as a final WTF – did you see the American Statistical Association have explicitly supported “climate change driven by CO2” position based on “rigourous scientific research”? Have I missed something since Wegman?:

    http://www.amstat.org/

    And the Society for Industrial and Applied Mathematics are signed up to – surely they know something about instrumentation, quality control and statistical significance?:

    http://www.siam.org/

    AAARRGGH!

    /rant off!

    On your point in paragraph 2 re: growth potential and ageing here is an abstract which sounds relevant:

    http://www.springerlink.com/content/8j71453650116753/

    And re: your closing comments about relative 19th Century temps. to the whole “reconstruction” and lack of 20thC up tick this seems relevant:

    http://www.springerlink.com/content/8j71453650116753/

    Apologies they are a bit drive by – they both need subscription.
    ******
    And any news on possible publication date for the Antarctic paper?

  4. Jeff Id said

    #2 hit CTRL ‘-‘

    #1 John, thanks. I put figure names so I could refer to the above. The red curve in Fig 9 is absolutely tempting for seeing temp. I see a nice warm spike in 1935ish and a strong gradual warming since 1975 but then after looking at the red line in Fig 10 you see well that warm spike was actually in 1925 so unless it was predicting the future…

    Also the 1960 range spikes with equal temps to 1996 and then you realize– this ain’t temp. Finally my last version where I RCS the whole thing (fig 11) shows entirely different peaks and valleys with a huge vertical spike on the end.

    Basically I just need to mess around with combinations of RCS curves until I get a bit of a gradual upslope with these squiggles, find some correlation factor and PUBLISH!! YEAH!

  5. Jeff Id said

    #3 Thanks also. The leading edge comment hurts a bit ’cause I’m just sorting spaghetti but I see you recognize that. Also, mom told me I’m nuts to keep doing this. It’s like banging your head on a wall. The stats are like a crossword and the vents are cathartic so I keep on going. Lately there hasn’t been time to read a lot of the general is or isn’t CO2 warming papers. There is a cool link though left by Kon Dealer which I checked out quickly.

    http://www.leif.org/EOS/2009GL039628-pip.pdf

  6. curious said

    PS re: instrumentation – http://chiefio.wordpress.com/2009/11/01/new-zealand-polynesian-polarphobia/#comments

  7. curious said

    Sorry Jeff – cross posted.

    Just to clarify – in no way was the “leading edge” comment a dig. I think what you (and loads of the other bloggers) do is amazing. I think we disagree on some stuff re: economy and energy but the need to get the spaghetti sorted is clear. The “is it worth it” is frustration – as you commented you have clear, meaningful, productive dialogue with (say) John Christy – wouldn’t it be amazing if this was the case across the board? That is what tipped me over the edge re: the splines thing – it seems the approach is: one gets an outlier data point which supports an a priori theory and so you locally soften up your curve fit to make sure you can take advantage of it. I call BS but I’m willing to be corrected.

    Thanks for kondealer’s link – will check it.

    And FWIW, I sort of see the leading edge of “climate science” akin to the leading edge of a bowl of blancmange dropped from a great height – who knows where it lies or what it means?! bw C🙂

  8. Steve Fitzpatrick said

    Very nice post.

    One thing that might make the spline correction curve (showing how trees start to grow faster after 200 years of life) more convincing would be to look at the behavior of trees that reached 200 years well before the 20th century (for example, reaching 200 years by 1800 or so). The growth rate of very old trees in the 20th century, if it is at all driven by temperature, is convoluted with both CO2 fertilization and an obvious selection bias toward very old trees by the Russians in the living population of trees, and this may be distorting the spline curve. If the same increase in growth after 200 years is seen in the 19th century and earlier, then that would pretty much prove the spline curve you propose is the correct one. Of course this doesn’t de-convolute CO2 and temperature in the 20th century growth rates. Growth effects of free-air CO2 enrichment on larch trees near the treeline would make an interesting study.

    Briffa has as much admitted that the last decade of the Yamal reconstruction was very doubtful (his words “treat with caution”) due to selection bias toward very old trees and very few samples, so I think it completely justified to simply ignore everything after ~1990 in Briffa’s reconstruction.

  9. Layman Lurker said

    Nice job Jeff. I’m not so sure about the weighting argument. It might take more analysis for each region such as that undertaken by JS at CA to explore the “year effects” and the “tree age effects” for each region separately to see how strong the individual growth functions are for each region.

    I thought it was just a matter of time before you pointed out that the same RCS was used in Briffa’s sensitivity as was used for the original Yamal. Aside from the unknown data selecting issues, this is the real crux of the matter.

  10. Jeff Id said

    #9 Thanks, I havent seen the JS individual region analysis (you are the master of keeping up on all the various posts) but in case it wasn’t clear fig’s 3 – 7 are individual weightings for Fig 9 and 10. 11 – 14 are a single weighting reconstruction – one size fit’s all – like the original Yamal and Roman’s post.

  11. Jeff Id said

    #8 That’s a good suggestion. I ran out of energy for it in this post but it’s probably worth the extra hour.

  12. chris y said

    Jeff, excellent post. Thank you for the buckets of work you put into this.

    In my opinion the need to cull tree ring records after the cores have been carefully selected in the field and analyzed is irrefutable evidence that tree rings cannot be used as a measure of past temperatures. The rest is all interesting details. I have yet to read any PaleoRorschachist provide a solid defense of the after-the-fact cherry-picking of records. Bizarrely, I have read of PaleoRorschachists bragging about it.

    I have a suggestion for the y axis label which you accurately show as “??”

    Perhaps something like “Arboreal karma”, or something related to astrological symbols.

  13. Kon Dealer said

    Nice work, Jeff.
    What this analysis shows me is 4 things:
    1) By changing the curve fit to the data, you change the outcome of the reconstruction. i.e. the conclusions are related to the method of analysis = “non-robust”
    2) One only gets the 3 sigma events (uptick) as the number of replicates fall below 20 = “non-robust”
    3) The reconstruction is highly sensitive to the choice of the end point = “non-robust”
    4) Old trees make up most, if not all, of the uptick. Old trees, by definition, are survivors, perhaps the ones most genetically able to make the most of a small amelioration in climate. Be it temperature, CO2 fertilisation, lack of competition from younger trees that died in the “Little Ice Age”, whatever? You tell me- how does Briffa KNOW it is temperature?-He can’t and what is more he can’t prove it.
    The fact of the matter is that these “survivors” are atypical and not representitive of the earlier parts of the chronology = “non-robust”.

  14. Layman Lurker said

    Just to be clear, JS applied his analysis on the original Yamal, not on Briffa’s sensitivity test, and it did not break it down by region. It’s on Roman’s “RCS – one size fits all” thread. JS does suggest in comment #392 that the model could break things down further – like looking at YAD only. Would the SE’s for each region be a good way to weight a RCS spline on the full data set?

  15. Kenneth Fritsch said

    Jeff ID, I have not thoroughly digested your analysis here, but from what I have, I would say I think you are on the right track. I must also comment that my understanding of the basis of the Yamal chronology and the influencial effects did not truly sink in until I did most of these analyses myself. I want to post my simple-minded analysis on Romanm’s Fits All thread in the near future.
    .

    Much of what I found was already revealed by the analyses of Romanm, Steve M, you and others, but the relevancy did not hit home until I did some for myself. All I want to do is put together my views on what are the important effects and why they are important.

    I would suggest looking at what NW posted at the Fits All thread at CA. Also I do not necessarily think that finding a more flexible fitting algorithm to accomodate the upward bend of the older tree ring age responses (deltas) will compensate for the accompanying noise levels. It has also been noted in the literature by Craig Loehle that the younger tree rings for larches (used in Yamal) are not very good climate indicators. We then have the situation where younger and older tree rings are problematic.

  16. jnorv said

    I ask again. There are living trees in California that are over 2000 years old and are still growing. Would not a tree ring sample from them be enlightening? It is a continuous record so there would be no need for the gyrations that the tree ring believers perform.

  17. Steve Fitzpatrick said

    #16,

    My understanding, from a very preliminary reading of dendro-climatology theory, is that only temperature stressed trees living near the cold limit of their range are expected to respond significantly to temperature variations. (Is this really the case? I have no idea.)

    The very old redwoods grow in relatively balmy northern California, and most likely the dendros think they would show nothing related to temperature.

  18. steven mosher said

    great post jeff. the right temp for correlation would be a seasonal value for JJA.

  19. Jeff Id said

    #16, Try and get a redwood coring permit🙂 I think 17 is probably right.

    #18 Thanks again.

  20. Arnost said

    #19 Try and get a redwood coring permit🙂

    Does not mean that there aren’t any… For example in:

    Multimillennial records of climatic variation in the Sierra Nevada inferred from tree rings” in Bulletin of the Ecological Society of America Jun 1995, Graumlich et al suggest that a dendochronological assessment on Sequoias has been made. From the Abstract (http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=95793):

    (I think a full print is here: http://www.wilderness.net/library/documents/Graumlich_3-x.pdf)

    Which brings me to why I’m delurking…

    There seems to be an a priori assumption that (as say #17) that “that only temperature stressed trees living near the cold limit of their range are expected to respond significantly to temperature variations”. This may well be – however, it does not preclude that these same trees do not respond to something else as well.

    I suggest (and this ain’t by all means an original point) that tree-ring chronologies used in the current paleo-temperature reconstructions may well be unsuitable unless the known/derived precipitation influence is regressed out (plus others).

    Case in point… One can see from Graumlich et al above, (at least in 1995) it seems well established that foxtails and bristlecones respond to precipitation. If (hypothetically) the tree-ring chronologies in Mann’s PC1 were influenced by increased precipitation during the training or calibration period, this may well reduce the true temperature signal in the reconstruction period. And of course, it just so happens that over this period there has been an increase in precipitation in Western / North Western US. See for example Fig 1 in: “Secular Trends of Precipitation Amount, Frequency, and Intensity in the United States” Thomas R. Karl and Richard W. Knight in the Bulletin of the American Meteorological Society back in October 1998 (full paper: http://www.umces.edu/president/hypoxiapubs/Karl%20Knight%201998.pdf).

    Back on topic… It is therefore probable that the larch also responds to precipitation… And of course it also happens that the Northern European / Russian precipitation trend is also up – at least since 1960. See fig 2 in http://www.meteo.ru/english/climate/extreme.php, (and if someone wants actual data, then possibly see here http://nsidc.org/data/g02170.html).

    (Note, there does seem “a lot” of precipitation in the Polar Urals – so maybe precipitation is not a factor in this case)

    Actually – it would be a fun exercise to cross reference each individual tree’s ring chronology to both temperature and precipitation! And here you could include sequoias that are better precipitation proxies to regress pre-instrumental variables and then maybe ENSO … Hmmm maybe when I retire. 🙂

    Food for thought.

    Cheers

    PS Sensational work you do here – Kudos

  21. Arnost said

    If this is a double post – sorry – maybe bad HTML…

    #19 “Try and get a redwood coring permit :)”

    Does not mean that there aren’t any… For example in:

    “Multimillennial records of climatic variation in the Sierra Nevada inferred from tree rings” in Bulletin of the Ecological Society of America Jun 1995, Graumlich et al suggest that a dendochronological assessment on Sequoias has been made. From the Abstract (http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=95793):

    Tree growth of foxtail pine (Pinus balfouriana; upper treeline, Sierra Nevada) is influenced by summer temperature and winter precipitation. Tree growth of giant sequoia (Sequoiadendron giganteum; west slope, Sierra Nevada) is influenced by short-term fluctuations in precipitation, especially extremely dry years, while growth of bristlecone pine (Pinus longaeva; lower forest border, White Mountains) reflects both short- and long-term variation in precipitation.”

    (I think a full print is here: http://www.wilderness.net/library/documents/Graumlich_3-x.pdf)

    Which brings me to why I’m delurking…

    There seems to be an a priori assumption that (as say #17) that “that only temperature stressed trees living near the cold limit of their range are expected to respond significantly to temperature variations”. This may well be – however, it does not preclude that these same trees do not respond to something else as well.

    I suggest (and this ain’t by all means an original point) that tree-ring chronologies used in the current paleo-temperature reconstructions may well be unsuitable unless the known/derived precipitation influence is regressed out (plus other influences).

    Case in point… One can see from the above Graumlich et al abstract, (at least in 1995) it seems well established that foxtails and bristlecones respond to precipitation. If (hypothetically) the tree-ring chronologies in Mann’s PC1 were influenced by increased precipitation during the training or calibration period, this may well reduce the true temperature signal in the reconstruction period. And of course, it just so happens that over this period there has been an increase in precipitation in Western / North Western US. See for example: “Secular Trends of Precipitation Amount, Frequency, and Intensity in the United States” Thomas R. Karl and Richard W. Knight in the Bulletin of the American Meteorological Society back in October 1998 (http://www.umces.edu/president/hypoxiapubs/Karl%20Knight%201998.pdf).

    Back on topic… It is therefore probable that the larch also responds to precipitation… And of course it also happens that the Northern European / Russian precipitation trend is also up – at least since 1960. See fig 2 in http://www.meteo.ru/english/climate/extreme.php, (and if someone wants actual data, then possibly see here http://nsidc.org/data/g02170.html).

    (Note: precipitation may not be a factor in this case as the Urals do get a substantial amount of precipitation in the summer months)

    Actually – it would be a fun exercise to cross reference each individual tree’s ring chronology to both temperature and precipitation! And here you could include sequoias that are better precipitation proxies to regress pre-instrumental variables and then maybe ENSO … Hmmm maybe when I retire. 🙂

    Food for thought.

    Cheers

    PS: You do great stuff here – Kudos.

  22. Jeff Id said

    Arnost, I had no idea coring would even be allowed. I agree on the assumptions, the one which really get’s me is the fact that CO2 is well known to increase growth. The whole point of global warming is correlation between CO2 and temps so if we were to actually see correlation with temp isn’t it possible that we’ve just accidentally reconstructed CO2??!!

  23. hunter said

    There is no reason to conclude anything except Briffa is playing the same game he has played for the past ~10 years: Manipulate for a desired result and hide the data.
    In business this is called ‘cover up’.

  24. Tom Fuller said

    Well done, sir.

    The Natural History Museum in London has a redwood trunk on one entire wall–sadly, I don’t recall when it was cut down.

  25. Jeff Id said

    Thanks Tom. It took a long time to dig through this data.

  26. qwerty said

    So does your analysis hinge on this point: “I’m critical of the exponential function method because it doesn’t recognize the potential for trees to increase in growth rate as they age. Since nothing I’ve read biologically defines that this is impossible it makes no sense to ignore the possibility unless you want older trees to curve upward after standardization.”

  27. Jeff Id said

    #26, Thats one of the differences right now, adding the new data doesn’t support the huge Yamal blade though either. Like many have pointed out though it doesn’t address the main criticisms of using trees as thermometers.

  28. qwerty said

    Does Briffa 2008 address use of an exponential function method?

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